Finite Group Databasebeta

← #29572527

Subgroups — #29572527

Order 15616.

690 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 107 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840374C211yesmaximal, characteristic
2#21445902C211yesmaximal, characteristic
3#21444351C214yesmaximal
4C61×#1820C211yesmaximal, characteristic
5#16840374—28no—
6C2×#14519777C411yescharacteristic, complemented
7C2×#14519770—22no—
8C2×#14519769C411yescharacteristic, complemented
9C2×#14519767—22no—
10C2×#14519756C2212yes—
11C2×C4×Dic122C412yescomplemented
12#16839726C2212yes—
13#16839531—22no—
14#16839498—28no—
15C2×#117×C61C2211yescharacteristic
16C61×#381C2212yes—
17C61×#342—22no—
18C22×Dic122C2×C412yes—
19C2×#11758194—22no—
20#14519777—24no—
21#14519770—216no—
22#14519769—24no—
23#14519767—216no—
24#14519756—24no—
25#14519756D418yes—
26C4×Dic122—24no—
27C4×Dic122—24no—
28C2×C4⋊Dic61—22no—
29C2×C4⋊Dic61C2×C412yes—
30C2×#11758196—22no—
31C2×#11758196—22no—
32C2×#11758196—22no—
33C2×C22⋊Dic61—22no—
34C2×C4×Dic61—22no—
35C22⋊#92418—24no—
36C23×C4×C61C2311yescharacteristic
37#117×C61—24no—
38C2×C4⋊C4×C61D412yes—
39C2×C22⋊C4×C61—22no—
40C2×C22⋊C4×C61D412yes—
41C2×C42×C61C2×C411yescharacteristic
42C2×C42×C61C2×C411yescharacteristic
43#95×C61—24no—
44C24×C61C2×D411yescharacteristic
45C22×C4×C61—42no—
46C22×C4×C61—42no—
47C22×C4×C61C22⋊C412yes—
48C22×C4×C61C22⋊C412yes—
49C22×C4×C61C22×C411yescharacteristic
50C22×C4×C61C2×D411yescharacteristic
51C22×C4×C61C2×D414yes—
52C22×Dic61—22no—
53C22×Dic61—22no—
54C2×Dic122—22no—
55C2×Dic122—44no—
56C2×Dic122C22⋊C414yes—
57C4⋊C4×C61—24no—
58C22⋊C4×C61—24no—
59C22⋊C4×C61—44no—
60C42×C61—22no—
61C42×C61—22no—
62C22⋊Dic61—28no—
63C4⋊Dic61—28no—
64C4⋊Dic61—22no—
65C4⋊Dic61C22⋊C414yes—
66#11758196—28no—
67#11758196—28no—
68#11758196—28no—
69C4×Dic61—28no—
70#11758194—44no—
71C2×#92418—42no—
72C23×C61—22no—
73C23×C61—42no—
74C23×C61C2×C22⋊C411yescharacteristic
75C23×C61#12012yes—
76C2×C4×C61—22no—
77C2×C4×C61—22no—
78C2×C4×C61—22no—
79C2×C4×C61—22no—
80C2×C4×C61—44no—
81C2×C4×C61—42no—
82C2×C4×C61—42no—
83C2×C4×C61—82no—
84C2×C4×C61—84no—
85C2×C4×C61C2×C22⋊C412yes—
86C2×C4×C61#12012yes—
87C2×Dic61—24no—
88C2×Dic61—24no—
89C2×Dic61—44no—
90C2×Dic61—44no—
91Dic122—44no—
92Dic122—44no—
93#92418—82no—
94#8521—611nomaximal, Sylow
95C22×C61—22no—
96C22×C61—24no—
97C22×C61—22no—
98C22×C61—82no—
99C22×C61#37911yescharacteristic
100C22×C61#41012yes—
101C4×C61—22no—
102C4×C61—22no—
103C4×C61—42no—
104C4×C61—42no—
105C4×C61—162no—
106Dic61—44no—
107Dic61—44no—
108C2×#541—611no—
109#1822—611no—
110#1820D6111yescharacteristic
111#1511—614no—
112C2×C61—22no—
113C2×C61—42no—
114C2×C61#178211yescharacteristic
115C2×C61#179612yes—
116#541—1228no—
117C2×C4⋊Q8—611no—
118C2×#126—1222no—
119C2×#125—611no—
120C2×#125—1222no—
121C2×C22⋊Q8—612no—
122C2×C4×Q8—612no—
123C2×#117C2×D6111yescharacteristic
124#407—612no—
125#381C2×D6112yes—
126#343—1222no—
127#342—22no—
128#328—1228no—
129C61#852111yescharacteristic, complemented, Sylow
130C22×Q8—612no—
131C23×C4C22×D6111yescharacteristic
132C2×M16—1222no—
133C4⋊Q8—1224no—
134#126—12216no—
135#125—1224no—
136#125—12216no—
137C22⋊Q8—618no—
138C22⋊Q8—1224no—
139C4×Q8—1224no—
140C4×Q8—1224no—
141#117—24no—
142C2×C4⋊C4—612no—
143C2×C4⋊C4—1222no—
144C2×C4⋊C4—1222no—
145C2×C4⋊C4—1222no—
146C2×C4⋊C4—1222no—
147C2×C4⋊C4D24412yes—
148C2×C22⋊C4—22no—
149C2×C22⋊C4—1222no—
150C2×C22⋊C4C22⋊D6112yes—
151C2×C42—1222no—
152C2×C42C4×D6111yescharacteristic
153C2×C42C4×D6111yescharacteristic
154C22⋊C8—1224no—
155#95—24no—
156C24C2×C22⋊D6111yescharacteristic
157C2×Q8—614no—
158C2×Q8—1222no—
159C2×Q8—2444no—
160C22×C4—42no—
161C22×C4—42no—
162C22×C4—1222no—
163C22×C4—1222no—
164C22×C4#1175819812yes—
165C22×C4#1175819812yes—
166C22×C4C2×C4×D6111yescharacteristic
167C22×C4C2×D24411yescharacteristic
168C22×C4D4×D6114yes—
169M16—2444no—
170C2×C8—2442no—
171C4⋊C4—24no—
172C4⋊C4—614no—
173C4⋊C4—1228no—
174C4⋊C4—1228no—
175C4⋊C4—1228no—
176C4⋊C4—1222no—
177C4⋊C4—1228no—
178C22⋊C4—24no—
179C22⋊C4—44no—
180C22⋊C4—1228no—
181C42—22no—
182C42—22no—
183C42—1228no—
184C23—22no—
185C23—42no—
186C23C2×#1175819811yescharacteristic
187C23#1451974512yes—
188Q8—2444no—
189Q8—2444no—
190C2×C4—22no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—44no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—82no—
198C2×C4—84no—
199C2×C4—1224no—
200C2×C4—1224no—
201C2×C4—2444no—
202C2×C4—2444no—
203C2×C4C22⋊C4×D6112yes—
204C2×C4#1451974712yes—
205C8—4882no—
206C22—22no—
207C22—22no—
208C22—24no—
209C22—82no—
210C22#1683961711yescharacteristic
211C22#1683973512yes—
212C4—22no—
213C4—22no—
214C4—42no—
215C4—42no—
216C4—162no—
217C4—2444no—
218C4—2444no—
219C2—22no—
220C2—42no—
221C2#2144573611yescharacteristic
222C2#2144579212yes—
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC23×C4C22×D61
FittingC61×#1820C2
socleC22×C61#379
prime radicalC24×C61C2×D4
cosocleC23×C4×C61C23
hypercenter#1820D61
hypocenterC61#8521
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1820

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
6. C2×#14519777C4semidirect
8. C2×#14519769C4semidirect
11. C2×C4×Dic122C4semidirect
129. C61#8521semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168403742
#214459022
#214443512
C61×#18202
#852161